Theorem xnegeq 8894

Description: Equality of two extended numbers with -_𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xnegeq	⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)

Proof of Theorem xnegeq

Step	Hyp	Ref	Expression
1		eqeq1 2087	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2		eqeq1 2087	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3		negeq 7301	. . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵)
4	2, 3	ifbieq2d 3373	. . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
5	1, 4	ifbieq2d 3373	. 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6		df-xneg 8843	. 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7		df-xneg 8843	. 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
8	5, 6, 7	3eqtr4g 2138	1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)

Syntax hints:   → wi 4   = wceq 1284  ifcif 3351  +∞cpnf 7150  -∞cmnf 7151  -cneg 7280  -_𝑒cxne 8840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-rab 2357  df-v 2603  df-un 2977  df-if 3352  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535  df-neg 7282  df-xneg 8843

This theorem is referenced by:  xnegcl  8899  xnegneg  8900  xneg11  8901  xltnegi  8902